Increasing likelihood ratio alone implies first-order domination due to the following (very) special case of the FKG inequality:

For any random variable $X$ and any two positive increasing functions $\phi,\psi$ we have $\mathsf{E} \, \phi(X) \psi(X) \ge \mathsf{E} \, \phi(X) \, \mathsf{E} \, \psi(X)$.

Now apply that to $X$ distributed according to $G$, and functions $\phi := \mathsf{1} [c,+\infty)$, $\psi := dF/dG$, and you will get exactly $G(c) \ge F(c)$.

Increasing likelihood ratio alone implies first-order domination due to the following (very) special case of the FKG inequality:

For any random variable $X$ and any two positive increasing functions $\phi,\psi$ we have $\mathsf{E} \, \phi(X) \psi(X) \ge \mathsf{E} \, \phi(X) \, \mathsf{E} \, \psi(X)$.

Apply that to $X$ distributed according to $G$, and functions $\phi := \mathsf{1} [c,+\infty)$, $\psi := dF/dG$, and you will get exactly $G(c) \ge F(c)$.